Which of the following is not a conclusion of the Central Limit Theorem?

• A. The standard deviation of the sampling distribution of the sample mean is equal to $\frac{\sigma }{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.
• B. The sample mean will always be exactly equal to the population mean for any sample size.
• C. The mean of the sampling distribution of the sample mean equals the population mean ($\mu$).
• D. The sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Answer: (B)

Verified step by step guidance

1. Recall the Central Limit Theorem (CLT) states that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution.
2. Identify the key conclusions of the CLT: (1) The mean of the sampling distribution of the sample mean equals the population mean, (2) The standard deviation of the sampling distribution (also called the standard error) is the population standard deviation divided by the square root of the sample size, and (3) The shape of the sampling distribution approaches normality as the sample size increases.
3. Examine each given statement and check if it aligns with these conclusions. The first, third, and fourth statements correspond directly to the CLT conclusions.
4. Focus on the statement 'The sample mean will always be exactly equal to the population mean for any sample size.' This is not a conclusion of the CLT because the sample mean is a random variable and can vary from sample to sample; it only approaches the population mean on average.
5. Therefore, the statement that the sample mean will always be exactly equal to the population mean is not a conclusion of the Central Limit Theorem.